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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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A Picone identity for variable exponent operators and applications

Rakesh Arora / Jacques Giacomoni / Guillaume Warnault
Published Online: 2019-05-16 | DOI: https://doi.org/10.1515/anona-2020-0003

Abstract

In this work, we establish a new Picone identity for anisotropic quasilinear operators, such as the p(x)-Laplacian defined as div(|∇ u|p(x)−2u). Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents. This new Picone identity can be also used to prove some accretivity property to a class of fast diffusion equations involving variable exponents. Using this, we prove for this class of parabolic equations a new weak comparison principle.

Keywords: parabolic problems with non standard growth; Picone identity for varible exponents; uniqueness; global behaviour

MSC 2010: 35A16; 35B51; 35K92; 35J92

1 Introduction and main results

The main aim of this paper is to prove a new version of the Picone identity involving quasilinear elliptic operators with variable exponent. The Picone identity is already known for homogeneous quasilinear elliptic as p-Laplacian with 1 < p < ∞.

In [1], M. Picone considers the homogeneous second order linear differential system

(a1(x)u)+a2(x)u=0(b1(x)v)+b2(x)v=0

and proved for differentiable functions u , v ≠ 0 the pointwise relation:

(uv(a1uvb1uv))=(b2a2)u2+(a1b1)u2+b1(uvuv)2(1.1)

and in [2], extended (1.1) to the Laplace operator, i.e. for differentiable functions u ≥ 0 , v > 0 one has

|u|2+u2v2|v|22uvu.v=|u|2(u2v).v0.(1.2)

In [3], Allegretto and Huang extended (1.2) to the p-Laplacian operator with 1 < p < ∞. Precisely, for differentiable functions v > 0 and u ≥ 0 we have

|u|p|v|p2v.(upvp1)0.

Picone identity plays an important role for proving qualitative properties of differential operators. In this regard, various attempts have been made to generalize Picone identity for different types of differential equations. At the same time, the study of differential equations and variational problems with variable exponents are getting more and more attention. Indeed, the mathematical problems related to nonstandard p(x)-growth conditions are connected to many different areas as the nonlinear elasticity theory and non-Newtonian fluids models (see [4, 5]). In particular the importance of investigating these kinds of problems lies in modelling various anisotropic features that occur in electrorheological fluids models, image restoration [6], filtration process in complex media, stratigraphy problems [7] and heterogeneous biological interactions [8]. The mathematical framework to deal with these problems are the generalized Orlicz Space Lp(x)(Ω) and the generalized Orlicz-Sobolev Space W1,p(x)(Ω). We refer to [9, 10, 11, 12, 13, 14] for the existence and regularity of minimizers in variational problems.

In [15, 16], several applications of Picone-type identity for p(⋅) = constant case have been obtained. This original identity is not further applicable for differential equations with p(x)-growth conditions. So, it is relevant to establish a new version of the Picone identity to include a large class of nonstandard p(x)-growth problems. In [5, 12, 17] convexity arguments to homogeneous functionals have been used to deal with quasilinear elliptic and parabolic problems with variable exponents. In the present paper, taking advantage of our new Picone pointwise identity, we give further applications in the context of elliptic and parabolic problems.

Before giving the statement of our main results, we first introduce notations and function spaces. Let Ω ⊂ ℝN, N ≥ 1. We recall some definitions of variable exponent Lebesgue and Sobolev spaces. Let 𝓟(Ω) be the set of all measurable function p : Ω → [1, ∞[ in N-dimensional Lebesgue measure. Define

ρp(u)=defΩ|u|p(x)dx.Lp(x)(Ω)={u:ΩR|u is measurable and ρp(u)<}

endowed with the norm

uLp(x)=inf{σ>0:ρp(uσ)1}.

The corresponding Sobolev space is defined as follows:

W1,p(x)(Ω)={uLp(x)(Ω):|u|Lp(x)(Ω)}

endowed with the norm

uW1,p(x)=uLp(x)+uLp(x)

and W01,p(x)(Ω)=W01,1(Ω)W1,p(x)(Ω)

In the sequel, we assume that Ω satisfies:

  • (Ω)

    For N = 1, Ω is a bounded open interval and for N ≥ 2, Ω is a bounded domain whose the boundary ∂ Ω is a compact manifold of class C1,γ for some γ ∈ (0, 1) and satisfies the interior sphere condition at every point of ∂ Ω.

Throughout the paper, we also assume that pC1(Ω). In addition, we suppose that

1<p=defminΩ¯p(x)p(x)p+=defmaxΩ¯p(x)<.

Then, W01,p(x)(Ω)=C0(Ω)¯W1,p(x)(Ω). We also recall some well-known properties on Lp(x) spaces (see [14]).

Proposition 1.1

Let pL(Ω). Then for any uLp(x)(Ω) we have:

  1. ρp(u/∥uLp(x)) = 1.

  2. uLp(x) → 0 if and only if ρp(u) → 0.

  3. Lpc(x)(Ω) is the dual space of Lp(x)(Ω) where we denote by pc the conjugate exponent of p defined as

    pc(x)=p(x)p(x)1.

Proposition 1.1 (i) implies that: if ∥uLp(x) ≥ 1,

uLp(x)pρp(u)uLp(x)p+(1.3)

and if ∥uLp(x) ≤ 1

uLp(x)p+ρp(u)uLp(x)p.(1.4)

Moreover, we have also the generalized Hölder inequality: for p measurable function in Ω, there exists a constant C = C(p+, p) ≥ 1 such that for any fLp(x)(Ω) and gLpc(x)(Ω)

Ω|f(x)g(x)|dxCfLp(x)gLpc(x).(1.5)

In Section 2, we prove the Picone identity for a general class of nonlinear operator. More precisely, we consider a continuous operator A : Ω × ℝN → ℝ such that (x, ξ) → A(x, ξ) is differentiable with respect to variable ξ and satisfies:

  • (A1)

    ξA(x, ξ) is positively p(x)-homogeneous i.e. A(x, t ξ) = tp(x) A(x, ξ), ∀ t ∈ ℝ+, ξ ∈ ℝN and a.e. xΩ.

  • (A2)]

    ξA(x, ξ) is strictly convex for any xΩ.

Remark 1.1

From the assumptions of A, we deduce A(x, ξ) > 0 for ξ ≠ 0 and for any xΩ.

By using the convexity and the p(x)-homogeneity of the operator A, we prove the following extension of the Picone identity:

Theorem 1.1

(Picone identity). Let A : Ω × ℝN → ℝ is a continuous and differentiable function satisfying (A1) and (A2). Let v0, vL(Ω) belonging to V˙+r=def{v:Ω(0,+)|v1rW01,p(x)(Ω)} for some r ≥ 1. Then

1p(x)ξA(x,v01/r),(vv0(r1)/r)Arp(x)(x,v1/r)A(p(x)r)p(x)(x,v01/r)

where 〈., .〉 is the inner scalar product and the above inequality is strict if r > 1 or vv0 ≢ Const > 0.

From the above Picone identity, we can show an extension of the famous Diaz-Saa inequality to the class of operators with variable exponent. This inequality is strongly linked to the strict convexity of some associated homogeneous energy type functional.

Theorem 1.2

(Diaz-Saa inequality). Let A : Ω × ℝN → ℝ is a continuous and differentiable function satisfying (A1) and (A2) and define a(x,ξ)=(ai(x,ξ))i=def1p(x)ξiA(x,ξ)i. Assume in addition that there exists Λ > 0 such that

aC1(Ω×(RN{0}))Nandi,j=1Nai(x,ξ)ξjΛ|ξ|p(x)2

for all (x, ξ) ∈ Ω × ℝN ∖ {0}. Then, we have in the sense of distributions, for any r ∈ [1, p]

Ω(div(a(x,w1))w1r1+div(a(x,w2))w2r1)(w1rw2r)dx0(1.6)

for any w1, w2W01,p(x)(Ω), positive in Ω such that w1w2,w2w1L(Ω). Moreover, if the equality occurs in (1.6), then w1/w2 is constant in Ω. If p(x) ≢ r in Ω then even w1 = w2 holds in Ω.

In sections 3, 4 and 5, we derive some applications of the new Picone identity. Precisely, we investigate the solvability of some boundary problems involving quasilinear elliptic operators with variable exponent.

In section 3, we consider the following nonlinear problem:

Δp(x)u+g(x,u)=f(x,u) in Ω;u>0 in Ω;u=0 on Ω.(1.7)

The extended Picone identity can be reformulated as in Lemma 3.1 below. Together with the strong maximum principle and elliptic regularity, this identity can be used to prove the uniqueness of weak solutions to elliptic equations as (1.7). In particular, we establish the following result:

Theorem 1.3

Let f, g : Ω × [0, ∞) → ℝ+ be defined as f(x, t) = h(x)tq(x)−1 and g(x, t) = l(x)ts(x)−1 with 1 ≤ q, sC(Ω) such that

  • q+ < p < s and q ≥ 1;

  • h, lL(Ω), positive functions such that xh(x)l(x)L(Ω).

Then, there exists a weak solution u to (1.7), i.e. u belongs to W01,p(x)(Ω) ∩ Ls(x)(Ω) and satisfies for any ϕW01,p(x)(Ω) ∩ Ls(x)(Ω):

Ω|u|p(x)2u.ϕdx=Ω(f(x,u)g(x,u))ϕdx.

Furthermore uC1,α(Ω) for some α ∈ (0, 1) and 0usq+max{hlL,1} a.e. in Ω.

Assume in addition that xl(x)h(x) belongs to L(Ω), then uCd0(Ω¯)+=def{vC0(Ω¯)|c1,c2R+:c1vdist(x,Ω)c2} and is the unique weak solution to (1.7).

Regarding the current literature, Theorem 1.3 does not require any subcritical growth condition for g to establish existence and uniqueness of the weak solution to (1.7).

In section 4, we study a nonlinear fast diffusion equation (F.D.E. for short) driven by p(x)-Laplacian. From the physical Fick’s law, the diffusion coefficient of our problem is then proportional to |∇ u(x, t)|p(x)−2. It naturally leads to investigate the following F.D.E. type problem:

q2q1t(u2q1)Δp(x)u=f(x,u)+h(t,x)uq1 in QT;u>0 in QT;u=0 on Γ;u(0,.)=u0 in Ω(1.8)

where q ∈ (1, p), QT = (0, T) × Ω and Γ = (0, T) × ∂ Ω for some T > 0. We suppose that hL(QT) and nonnegative. The assumptions on f are given by

  • (f1)

    f : Ω × ℝ+ → ℝ+ is a function such that f(x, 0) = 0 for all xΩ and f ≢ 0;

  • (f2)

    lims0+f(x,s)s2q1=+ uniformly in x;

  • (f3)

    for any xΩ, sf(x,s)sq1 is nonincreasing in ℝ+ ∖ {0}.

Remark 1.2

Conditions (f1) and (f3) imply there exist positive constant C1, C2 such that for any (x, s) ∈ Ω × ℝ+:

0f(x,s)C1+C2sq1,

i.e. f has a strict subhomogeneous growth.

We set 𝓡 the operator defined by Rv=Δp(x)(v1/q)v(q1)/qf(x,v1/q)v(q1)/q and the associated domain

D(R)={v:Ω(0,):v1/qW01,p(x)(Ω),vL2(Ω),RvL2(Ω)}.

Note that 𝓓(𝓡) contains for instance solutions to (4.18). One can also easily check that solutions to (4.19) belong to 𝓓(𝓡)L2(Ω). In the sequel, we denote X+=def{xX|x0} the associated positive cone of a given real vector space X.

In order to establish existence and properties of weak solutions to (1.8), we investigate the following related parabolic problem:

vq1t(vq)Δp(x)v=h(t,x)vq1+f(x,v) in QT;v>0 in QT;v=0 on Γ;v(0,.)=v0(x)>0 in Ω.(1.9)

The notion of weak solution for (1.9) is given as follows:

Definition 1.1

A weak solution to (1.9) is any positive function

vL(0, T; W01,p(x)(Ω)) ∩ L(QT) ∩ C(0, T; Lr(Ω)) for any r ≥ 1 such that t(vq) ∈ L2(QT) and for any ϕC0(QT) satisfies

0TΩt(vq)vq1ϕdxdt+0TΩ|v|p(x)2v.ϕdxdt=0TΩh(t,x)vq1ϕdxdt+0TΩf(x,v)ϕdxdt.(1.10)

Concerning (1.9), we prove the following results:

Theorem 1.4

Let T > 0, v0Cd0(Ω¯)+W01,p(x)(Ω). In addition, there exists h0L(Ω), h0 ≢ 0 and h(t, x) ≥ h0(x) ≥ 0 for a.e xΩ, for a.e. t ≥ 0. Assume in addition qmin{N2+1,p} and f satisfies (f1) − (f3). Then there exists a weak solution to (1.9).

Based on the accretivity of 𝓡 with domain 𝓓(𝓡), we show the following result providing a contraction property for weak solutions to (1.9) under suitable conditions on initial data:

Theorem 1.5

Let v1 and v2 are weak solutions of (1.9) with initial data u0, v0Cd0(Ω¯)+W01,p(x)(Ω) and such that u0q,v0qD(R)¯L2(Ω) and h, gL(QT), such that hh0, gg0 with h0, g0 as in Theorem 1.4. Then, for any 0 ≤ tT,

(v1q(t)v2q(t))+L2(u0qv0q)+L2+0t(h(s)g(s))+L2ds.(1.11)

Furthermore, using a similar approach as in [8], we consider for ϵ > 0 the perturbed operator 𝓡ϵ v = Δp(x)(v1/q)(v+ϵ)(q1)/qf(x,v1/q)(v+ϵ)(q1)/q. If p ≥ 2, we can prove (as in Proposition 2.6 in [8]) that

D(Rϵ)¯L2(Ω)V˙+qCdq0(Ω¯)+.

Arguing as in Theorem 1.5 with the operator 𝓡ϵ instead of 𝓡 and passing to the limit as ϵ → 0+, we get:

Corollary 1.1

Assume p ≥ 2. Let v1 and v2 are weak solutions of (1.9) with initial data u0, v0Cd0(Ω¯)+W01,p(x)(Ω). Then Theorem 1.5 holds.

From Theorem 1.5, we derive the following comparison principle from which uniqueness of the weak solution to problem (1.9) follows:

Corollary 1.2

Let u and v are the weak solutions of (1.9) with initial data u0, v0 satisfying conditions in Theorem 1.5 or Corollary 1.1. Assume u0v0 and h, gL(QT) , h0L(Ω) such that and 0 < h0hg. Then uv.

Remark 1.3

If vL(QT)+ then from Proposition 9.5 in [18] we obtain q2q1 t(v2q−1) = vq−1 t(vq) = q v2q−2 t v in weak sense.

Remark 1.4

From Theorem 1.5, we can derive stabilization results for the evolution equation (1.8) in Lq(Ω) with q ∈ [2, ∞) (see [12] in this regard).

From the above remark, under assumptions given in Theorem 1.4, we obtain the existence of weak solutions to (1.8) satisfying the monotonicity properties in Theorem 1.5 and Corollaries 1.1, 1.2. We highlight that in our knowledge there is no result available in the current literature about F.D.E. with variable exponent. In this regard our results are completely new.

In the previous applications, the condition (A1) plays a crucial role to get suitable convexity property of energy functionals. In section 5, we study a quasilinear elliptic problem where this condition is not satisfied. Precisely, given ϵ > 0, we study the following nonhomogeneous quasilinear elliptic problem:

div((|u|2+ϵu2)p(x)22u)(|u|2+ϵu2)p(x)22ϵu=g(x,u) in Ω;u=0 on Ω;u>0 in Ω(1.12)

where g satisfies (f1) and () for some m ∈ [1, p]:

  • (g̃)

    For any xΩ, sg(x,s)sm1 is decreasing in ℝ+ ∖ {0} and a.e. in Ω.

Then we prove the following result:

Theorem 1.6

Assume that g satisfies (f1) and (). Then for any ϵ, (1.12) admits one and only one positive weak solution. Furthermore, uC1(Ω), u > 0 in Ω and un < 0 on ∂ Ω.

To get the uniqueness result contained in Theorem 1.6, we exploit the hidden convexity property of the associated energy functional in the interior of positive cone of C1(Ω).

2 Picone identity and Diaz-Saa inequality

2.1 Picone identity

First we recall the notion of strict ray-convexity.

Definition 2.1

Let X be a real vector space. Let be a non empty cone in X. A function J : → ℝ is ray-strictly convex if for all v1, v2 and for all θ ∈ (0, 1)

J((1θ)v1+θv2)(1θ)J(v1)+θJ(v2)

where the inequality is always strict unless v1 = Cv2 for some C > 0.

Then we have the following result:

Proposition 2.1

Let A satisfying (A1) and (A2) and let r ≥ 1. Then, for any xΩ the map ξNr(x, ξ) =def A(x, ξ)r/p(x) is positively r-homogeneous and ray-strictly convex. For r > 1, ξNr(x, ξ) is even strictly convex.

Proof

We begin by the case r = 1. For any t ∈ ℝ+, we have N1(x, t ξ) = t N1(x, ξ). Furthermore,

A(x,(1t)ξ1+tξ2)(1t)A(x,ξ1)+tA(x,ξ2)max{A(x,ξ1),A(x,ξ2)}

for any xΩ, ξ1, ξ2 ∈ ℝN and t ∈ [0, 1]. Therefore

N1(x,(1t)ξ1+tξ2)max{N1(x,ξ1),N1(x,ξ2)}(2.1)

and this inequality is always strict unless ξ1 = λ ξ2, for some λ > 0.

Now we prove that N1 is subadditive.

Without loss of generality, we can assume that ξ1 ≠ 0 and ξ2 ≠ 0 . Then we have N1(x, ξ1) > 0 and N1(x, ξ2) > 0. Therefore, from (2.1) and 1-homogeneity of N1(x, ξ) we obtain for any t ∈ (0, 1):

N1(x,(1t)ξ1N1(x,ξ1)+tξ2N1(x,ξ2))1.

We now fix t such that

1tN1(x,ξ1)=tN1(x,ξ2)i.e.t=N1(x,ξ2)N1(x,ξ1)+N1(x,ξ2)1.

Then we get

N1(x,ξ1+ξ2N1(x,ξ1)+N1(x,ξ2))1

and by 1-homogeneity of N1, we obtain

N1(x,ξ1+ξ2)N1(x,ξ1)+N1(x,ξ2),i.e.N1is subadditive.

Finally for t ∈ (0, 1), ξ1λ ξ2, ∀ λ > 0

N1(x,(1t)ξ1+tξ2)<N1(x,(1t)ξ1)+N1(x,tξ2)=(1t)N1(x,ξ1)+tN1(x,ξ2).

This proves that ξN1(x, ξ) is ray-strictly convex. Now consider the case r > 1. Since for any xΩ, ξNr1/r(x, ξ) = N1(x, ξ) is ray-strictly convex and thanks to the strict convexity of ttr on ℝ+, we deduce that ξNr(x, ξ) = N1r(x, ξ) is strictly convex when r > 1. □

From Proposition 2.1 and from the r-homogeneity of Nr, we easily deduce the following convexity property of the energy functional:

Proposition 2.2

Under hypothesis of Proposition 2.1 and assume in addition A is continuous on Ω × ℝN. Then, for 1 ≤ r < p:

V˙+rL(Ω)vΩA(x,(v1/r))dx

is ray-strictly convex (if r > 1, it is even strictly convex).

Proof

We know that ξNr(x, ξ) = Ar/p(x)(x, ξ) is r-positively homogeneous and strictly convex if r > 1 and for r = 1 this function is ray-strictly convex. For v1, v2V˙+r and θ ∈ (0, 1) define v = (1 − θ) v1 + θ v2 and we get

Nr(x,vv)(1θ)v1vNr(x,v1v1)+θv2vNr(x,v2v2).

By homogeneity,

Nr(x,(v1/r))(1θ)Nr(x,(v11/r))+θNr(x,(v21/r))

and equality holds if and only if v1 = λ v2 for some λ > 0. Using the convexity of ttp(x)/r for 1 ≤ r < p we obtain

ΩA(x,v1/r)dx(1θ)ΩA(x,v11/r)dx+θΩA(x,v21/r)dx.

Moreover, if p(x) ≠ r equality holds if and only if v1 = v2. □

From Proposition 2.1, we deduce the proof of Picone identity.

Proof of Theorem 1.1

Firstly, we deal with the case r > 1. Then from Proposition 2.1, for any xΩ the function ξNr(x, ξ) = A(x, ξ)r/p(x) is strictly convex. Let ξ , ξ0 ∈ ℝN ∖ {0} such that ξξ0 then

Nr(x,ξ)Nr(x,ξ0)>ξNr(x,ξ0),ξξ0.

Setting a~(x,ξ)=1rξNr(x,ξ), we obtain:

Nr(x,ξ)a~(x,ξ0),ξ0>ra~(x,ξ0),ξξ0.

Let v, v0 > 0 and replacing ξ, ξ0 by ξ/v and ξ0/ v0 respectively in the above expression, we get

Nr(x,ξv)>ra~(x,ξ0v0),ξvr1rξ0v0.

Taking ξ = ∇ v and ξ0 = ∇ v0 and using (r − 1)-homogeneity of ã(x, .),

N(x,vrv(r1)/r)>1v0(r1)/ra~(x,v0rv0(r1)/r),vr1rv0v0v

where the inequality is strict unless vv=v0v0.

Since v1/r, v01/rW1,p(x)(Ω) ∩ L(Ω), we can write

(v1/r)=vrv(r1)/rand(vv0(r1)/r)=1v0(r1)/r(vr1rv0v0v)

and we obtain

N(x,v1/r)>a~(x,v01/r),(vv0(r1)/r).(2.2)

We have

a~(x,v01/r)=1rξN(x,v01/r)=1rξAr/p(x)(x,v01/r)=1p(x)ξA(x,v01/r)Arp(x)p(x)(x,v01/r)

and by replacing in (2.2) we obtain

Arp(x)(x,v1/r)Ap(x)rp(x)(x,v01/r)>1p(x)ξA(x,v01/r),(vv0(r1)/r).

Now we deal with the case r = 1. Let ξ , ξ0 ∈ ℝN ∖ {0} such that for any λ > 0, ξλ ξ0. Then, from Proposition 2.1, we have that

N(x,ξ)N(x,ξ0)ξN(x,ξ0),ξξ0.

Taking ξ = ∇ v and ξ0 = ∇ v0, we deduce

N(x,v)N(x,v0)ξN(x,v0),(vv0)

and

A1p(x)(x,v)Ap(x)1p(x)(x,v0)1p(x)ξA(x,v0),v

for any xΩ and the inequality is strict unless v = λ v0 for some λ > 0. □

The Picone identity also holds for anisotropic operators of the following type:

i=1Ni(bi(x,iu))=i=1Nxibix,uxi.

Precisely we have:

Corollary 2.1

Let B : Ω × ℝ → ℝN is a continuous and differentiable function such that B(x, s) = (Bi(x, s))i=1,2,…N satisfying for any i, for any xΩ, the map sBi(x, s) is pi(x)-homogeneous and strictly convex with 1 < pipi()pi+ < ∞. For any i, we define bi(x, s) = 1pi(x) s Bi (x, s). Then, for v, v0V˙+rL(Ω), we have

i=1Nbi(x,xi(v01/r))xivv0r1ri=1NBirpi(x)x,xi(v1/r)Bipi(x)rpi(x)x,xi(v01/r).

Proof

By taking A(x, s) = Bi(x, s) in Theorem 1.1, we obtain ∀ i ∈ {1, 2, …, N}

1pi(x)sBi(x,xi(v01/r)).xivv0r1rBirpi(x)(x,xi(v1/r)).Bipi(x)rpi(x)(x,xi(v01/r))

for all v, v0V˙+rL(Ω) and i = 1, 2, …, N.

Then by summing the expression over i = 1, 2, …, N, we obtain

i=1Nbi(x,xi(v01/r)).xivv0r1ri=1NBirpi(x)(x,xi(v1/r)).Bipi(x)rpi(x)(x,xi(v01/r)).

2.2 An extension of the Diaz-Saa inequality

We prove the first application of Picone identity.

Proof of Theorem 1.2

The Picone identity implies

Ar/p(x)(x,w1)A(p(x)r)/p(x)(x,w2)a(x,w2).(w1rw2r1).

Using the Young inequality for r ∈ [1, p], we get

rp(x)(A(x,w1)A(x,w2))+A(x,w2)a(x,w2).(w1rw2r1).

Noting that for any ξ ∈ ℝN, A(x, ξ) = a(x, ξ). ξ, we deduce

a(x,w2).(w2w1rw2r1)dxrp(x)(A(x,w2)A(x,w1)).(2.3)

Commuting w1 and w2, we have

a(x,w1).(w1w2rw1r1)rp(x)(A(x,w1)A(x,w2)).(2.4)

Summing (2.3) and (2.4) and integrating over Ω yield

Ωa(x,w1).(w1rw2rw1r1)dx+Ωa(x,w2).(w2rw1rw2r1)0.

The rest of the proof is the consequence of Proposition 2.2. □

Diaz-Saa inequality also holds for anisotropic operators. Here we require that ξBi(x, ξ) is pi(x)-homogeneous and strictly convex and bi(x, ξ) = 1pi(x) i Bi (x, ξ) where r ∈ ℝ, 1 ≤ r ≤ mini=1,2,…,N {(pi)}.

Corollary 2.2

Under the assumptions of Corollary 2.1 and in addition that there exist Λ > 0 such that for each i, bis(x,s)Λ|s|p(x)2. Then we have in the sense of distributions, for r ∈ [1, mini {(pi)}] and v, v0V˙+rL(Ω):

i=1NΩ(xi(bi(x,xiv))vr1(x)+xi(bi(x,xiv0))v0r1(x))(vrv0r)dx0

Proof

We apply Theorem 1.2. For A = Bi : Ω × ℝ → ℝ and by replacing ∇ by xi. □

3 Application of Picone identity to quasilinear elliptic equations

The aim of this section is to establish Theorem 1.3.

3.1 Preliminary results

The first lemma is the Picone identity in the context of the p(x)-Laplacian operator.

Lemma 3.1

Let r ∈ [1, p] and u, vW01,p(x) (Ω) ∩ L(Ω) two positive functions. Then for any xΩ

|u|p(x)+|v|p(x)|v|p(x)2v.(urvr1)+|u|p(x)2u.(vrur1).

Following the proof of Theorem 1.1 in [19], we first prove the following comparison principle:

Lemma 3.2

Let λ ≥ 0 and u, vW01,p(x)(Ω) ∩ Lα(x)(Ω) two nonnegative functions for some function α ∈ 𝓟(Ω) satisfying 1 < αα+ < ∞. Assume for any ϕW01,p(x)(Ω), ϕ ≥ 0:

Ω|u|p(x)2u.ϕ+uα(x)1χuλϕdxΩ|v|p(x)2v.ϕ+vα(x)1χvλϕdx

where

χvλ(x)=1ifλv<;0if0v<λ,

and uv a.e. in ∂ Ω. Then uv a.e. in Ω.

Proof

Let ϕ = (vu)+W01,p(x)(Ω) and Ω1 = {xΩ : u(x) < v(x)}. Then

0Ω1(|u|p(x)2u|v|p(x)2v).(uv)dxΩ1(uα(x)1χuλvα(x)1χvλ)(uv)dx0

from which we obtain uv a.e. in Ω. □

Using lemma 3.2, we show the following strong maximum principle:

Lemma 3.3

Let h, lL(Ω) be nonnegative functions, h > 0 and k : Ω × ℝ+ → ℝ+. Let α, β ∈ 𝓟(Ω) be two functions such that 1 < ββ+ < αα+ < ∞. Let uC1(Ω) be nonnegative and a nontrivial solution to

Δp(x)u+l(x)uα(x)1=h(x)uβ(x)1+k(x,u)inΩ;u=0onΩ.(3.1)

Assume in addition either

  • (c1)

    lhL(Ω)

    or

  • (c2)

    k : Ω × ℝ+ → ℝ+ satisfying lim inft0+k(x,t)tα(x)1>lL uniformly in x.

Then u is positive in Ω.

Proof

We follow the idea of the proof of Theorem 1.1 in [19]. For the reader’s convenience we have included the detailed proof. We rewrite our equation (3.1) under condition (c1) as follows:

Δp(x)u+l(x)uα(x)1χuλh(x)uβ(x)1(1χuλ)1l(x)h(x)uα(x)β(x),

since lhL(Ω), we choose λ ∈ (0, 1) small enough such that for any u(x) ≤ λ, we have 1 − l(x)h(x)uα(x)β(x)1lhL(Ω)λαβ+0.

Assuming condition (c2), we have

Δp(x)u+l(x)uα(x)1χuλk(x,u)(1χuλ)l(x)uα(x)1.

We choose λ small enough such that for any u(x) ≤ λ, we have k(x, u) − l(x) uα(x)−1 ≥ 0. Hence under both conditions, we get for any xΩ,

Δp(x)u+l(x)uα(x)1χuλ0.

Suppose that there exists x1 such that u(x1) = 0 then using the fact that u is nontrivial, we can find x2Ω and a ball B(x2, 2C) in Ω such that x1 B(x2, 2C) and u > 0 in B(x2, 2C).

Let a = ∈ f{u(x) : |xx2| = C} then a > 0 and choosing x2 close enough to x1 such that 0 < a < λ and ∇ u(x1) = 0 since u(x1) = 0.

Denote the annulus P = {xΩ : C < |xx2| < 2C}. We define p1 = p(x1), M = sup{|∇ p(x)| : xP}, b = 8M + 2, l1 = −b ln aC+2(N1)C and

j(t)=ael1Cp111(el1tp111)t[0,C].

We have

aCel1Cp11<j(0)j(t)j(C)<aCel1Cp11

and then

(aC)3j(t)1t[0,C].(3.2)

We choose C < 1 and using ∇ u(x1) = 0, aC < 1 small enough such that for any xP

p(x)1p1112.(3.3)

Without loss of generality we can take x2 = 0 and we set r = |xx2| = |x|, t = 2Cr. For t ∈ [0, C] and r ∈ [C, 2C], denote w(r) = j(2Cr) = j(t), then

w(r)=j(t),w(t)=j(t).

From (3.2) and (3.3), we obtain

div(|w|p(x)2w)=(p(x)1))(j(t))p(x)2j(t)N1r(j(t))p(x)1(j(t))p(x)1ln(j(t))i=1npxi.xir(j(t))p(x)1(12l1+Mln(j(t))N1r)ln(aC)(j(t))p(x)10.

Since j(t) < a < λ , we deduce

div(|w|p(x)2w)+wα(x)1χwλ0.

On ∂ P, w(C) = j(C) = au(x) and w(2C) = j(0) = 0 ≤ u(x). Then by Lemma 3.2, we obtain wu on P. Finally,

lims0+u(x1+s(x2x1))u(x1)slims0+w(x1+s(x2x1))w(x1)s=j(0)>0

which contradicts ∇ u(x1) = 0. Therefore, u > 0 in Ω. □

Remark 3.1

Conditions (c1) and (c2) can be replaced by the condition that there exists t0 such that h(x)tβ(x)−1 + k(x, t) − l(x) tα(x)−1 ≥ 0 for all 0 < t < t0 and xΩ.

Lemma 3.4

Under the same conditions of h, l, k as in Lemma 3.3, let uC1(Ω) be the nonnegative and nontrivial solution of (3.1), x1∂ Ω , u(x1) = 0 and Ω satisfies the interior ball condition at x1, then un(x1) < 0 where n⃗ is the outward unit normal vector at x1.

Proof

Choose C > 0 small enough such that B(x2, 2C) ⊂ Ω, x1 B(x2, 2C). Then x2 = x1 + 2 C n⃗, where n⃗ is the outward normal at x1. Denote P = {xΩ : C < |xx2| < 2C} and by choosing a such that 0 < a < λ, then by Lemma 3.3, there exist a subsolution wC1(P) ∩ C2(P) of (3.1) in P and w satisfies wu in P with w(x1) = 0, wn(x1) < 0. Hence, we get un(x1)wn(x1)<0. □

3.2 Proof of Theorem 1.3

Proof of Theorem 1.3

We perform the proof along five steps. First we introduce notations. Define F, G : Ω × ℝ → ℝ+ as follows:

F(x,t)=h(x)q(x)tq(x) if 0t<;0 if <t<0,

and

G(x,t)=l(x)s(x)ts(x) if 0t<;0 if <t<0.

We also extend the domain of f and g to all Ω × ℝ by setting

f(x,t)=Ft(x,t)=0andg(x,t)=Gt(x,t)=0for(x,t)Ω×(,0).

Define the energy functional 𝓔 : W01,p(x)(Ω) ∩ Ls(x)(Ω) → ℝ by

E(u)=Ω|u|p(x)p(x)dx+ΩG(x,u(x))dxΩF(x,u(x))dx.(3.4)

  • Step 1

    Existence of a global minimizer

    Since W01,p(x)(Ω) ↪ Lq(x)(Ω) (see Theorem 3.3.1 and Theorem 8.2.4 in [9]), the functional 𝓔 is well-defined for every function uW01,p(x)(Ω) ∩ Ls(x)(Ω).

    For uW01,p(x) large enough: by (1.3) or (1.4)

    E(u)Ω|u|p(x)p(x)Ωh(x)q(x)|u|q(x)1puLp(x)pCρq(u)1puW01,p(x)pCuW01,p(x)q~

    where q~=q if uLp(x)1q+ if uLp(x)>1 Since p > q+, this implies

    E(u)asuW01,p(x)+.

    We argue similarly when |u|Ls(x) → ∞ and we deduce 𝓔 is coercive. The continuity of 𝓔 on W01,p(x)(Ω) ∩ Ls(x)(Ω) is given by Theorem 3.2.8 and 3.2.9 of [9]. Hence we get the existence of at least one global minimizer, say u0, to (3.4).

  • Step 2

    Claim: u0 ≥ 0 and u0 ≢ 0

    Since u0 is a global minimizer of 𝓔 then 𝓔(u0+) ≥ 𝓔(u0) where u0+ = max{u0, 0} ∈ W01,p(x)(Ω). Set Ω = {xΩ : u0(x) < 0}. We have

    E(u0)=Ω|u0|p(x)p(x)dx+ΩG(x,u0(x))dxΩF(x,u0(x))dx=E(u0+)+Ω|u0|p(x)p(x)dx

    which implies Ω|u0|p(x)p(x)=0 i.e. ∇u0(x) = 0 a.e. in Ω then by (1.3) and (1.4) we have u0 = 0 a.e in Ω. This implies that u0 ≥ 0.

    In order to show that u0 ≢ 0 in Ω, we construct a function v in W01,p(x)(Ω)∩ L(Ω) such that 𝓔(v) < 0 = 𝓔(0). Precisely, consider v = where ϕCc1(Ω), ϕ ≥ 0, ϕ ≢ 0 in Ω and for 0 < t ≤ 1 small enough, we have

    E(v)tq+(c1tpq++c2tsq+c3)

    where for any i ∈ {1, 2, 3}, ci are suitable constants independent of t. Hence, choosing t small enough the right-hand side is negative and we conclude that 𝓔() < 0 = 𝓔(0) which implies u0 ≢ 0.

  • Step 3

    u0 satisfies the equation in (1.7)

    Since u0 is a global minimizer and 𝓔 is C1 on W01,p(x)(Ω) ∩ Ls(x)(Ω), then for any ϕW01,p(x)(Ω) ∩ Ls(x)(Ω), we have

    E(u0),ϕ=Ω|u0|p(x)2u0.ϕdxΩf(x,u0)ϕdx+Ωg(x,u0)ϕdx=0.

  • Step 4

    Regularity and positivity of weak solutions

    First we prove that all nonnegative weak solutions of (1.7) belongs to L(Ω) which yields C1, α(Ω) regularity.

    Let K(x, t) = h(x) tq(x)–1l(x) ts(x)–1 and Λ=defmax{hlL,1}1/(sq+)

    Then it is not difficult to show that for any tΛ, K(x, t) ≤ 0.

    Let u be a nonnegative function satisfying weakly the equation in (1.7). Then for any ϕW01,p(x)(Ω) ∩ Ls(x)(Ω),

    Ω|u|p(x)2u.ϕdx=Ω(h(x)uq(x)1l(x)us(x)1)ϕ(x)dx.

    Taking the testing function ϕ(x) = (uΛ)+, we get

    Ω|(uΛ)+|p(x)0.

    By using (1.4), we deduce (uΛ)+W01,p(x)=0 which implies u(x) ≤ Λ.

    From Theorem 1.2 in [10], we get uC1,α(Ω) for some α ∈ (0, 1).

    Furthermore assuming xl(x)h(x) belongs to L(Ω), Lemma 3.3 yields u > 0 in Ω.

  • Step 5

    Uniqueness of the positive solution of (1.7)

    Let u, v be two positive solutions of (1.7). Thus for any ϕ, ϕ̃W01,p(x)(Ω) ∩ Ls(x)(Ω),

    Ω|u|p(x)2u.ϕdx=Ω(h(x)uq(x)1l(x)us(x)1)ϕ(x)dx

    and

    Ω|v|p(x)2v.ϕ~dx=Ω(h(x)vq(x)1l(x)vs(x)1)ϕ~(x)dx.

    By the previous steps, u and v belong to C1(Ω) and Lemma 3.4 implies u, vCd0(Ω¯)+. Hence taking the testing functions as ϕ=(upvp)+up1andϕ~=(vpup)vp1W01,p(x)(Ω) (with the following notation t=def max{0, –t}) and from Lemma 3.1 we obtain

    0{u>v}(|u|p(x)2u|v|p(x)2v).(uv)dx={u>v}h(x)(uq(x)pvq(x)p)(upvp)dx+{u>v}l(x)(vs(x)pus(x)p)(upvp)dx.

    Since q+ps, the both terms in right-hand side are nonpositive. This implies v(x) ≥ u(x) a.e in Ω.

    Finally reversing the role of u and v, we get u = v.□

Remark 3.2

Theorem 1.3 still holds when the condition lhL(Ω) is replaced by p+ < s and using strong maximum principle in [19].

4 Application to Fast diffusion equations

In this section, we establish Theorems 1.4 and 1.5. To this aim, we use a time semi-discretization method associated to (1.9). With the help of accurate energy estimates about the related quasilinear elliptic equation and passing to the limit as the discretization parameter goes to 0, we prove the existence and the properties of weak solutions to (1.8). In the subsection below, we study the associated elliptic problem.

4.1 Study of the quasilinear elliptic problem associated to F.D.E.

Consider the following problem

v2q1λΔp(x)v=h0(x)vq1+λf(x,v) in Ω;v>0 in Ω;v=0 on Ω.(4.1)

Assume h0L(Ω)+ and f satisfies (f1)-(f3). Then from (f3), we have (f0) : lims+f(x,s)sp1=0 uniformly in xΩ. Therefore, for any ϵ > 0, there exists a positive constant Cϵ such that for any (x, s) ∈ Ω × ℝ+:

0f(x,s)Cϵ+ϵsp1.(4.2)

We have the following preliminary result about (4.1):

Theorem 4.1

Let λ > 0, q ∈ (1, p], f : Ω × ℝ+ → ℝ+ satisfying (f0) and (f1) and h0L(Ω)+. Then there exists a weak solution vC1(Ω) to (4.1), i.e. for any ϕW =def W01,p(x)(Ω) ∩ L2q(Ω)

Ωv2q1ϕdx+λΩ|v|p(x)2v.ϕdx=Ωh0vq1ϕdx+λΩf(x,v)ϕdx.(4.3)

In addition, if (f2) and (f3) hold then vCd0(Ω¯)+. Moreover if v1, v2Cd0(Ω¯)+ are two weak solutions to (4.1) corresponding to h0 = h1, h2L(Ω)+ respectively, then we have

(v1qv2q)+L2(h1h2)+L2.(4.4)

Remark 4.1

(4.4) implies the uniqueness of the weak solution to (4.1) in Cd0(Ω¯)+.

Proof

We perform the proof into several steps.

  • Step 1

    Existence of a weak solution

    Consider the energy functional 𝓙 defined on W equipped with .W=.W01,p(x)+.L2q

    J(v)=12qΩv2qdx+λΩ|v|p(x)p(x)dx1qΩh0D(v)dxλΩF(x,v)dx(4.5)

    where

    D(t)=tq if 0t<;0 if <t<0,andF(x,t)=0tf(x,s)ds if 0t<;0 if <t<0.

    We also extend the domain of f to all of Ω × ℝ by setting f(x,t)=Ft(x,t)=0 for (x, t) ∈ Ω × (–∞, 0). From (4.2), Hölder inequality (1.5) and since W01,p(x)Lp(Ω), we obtain

    J(v)12qvL2q2q+λvW01,p(x)p1qh0L2vL2qqλCϵΩ|v|dxλϵpΩ|v|pdx1qvL2qq(12vL2qqh0L2)+λvW01,p(x)((1ϵ)vW01,p(x)p1C~).

    Then by choosing ϵ small enough we conclude the coercivity of 𝓙 on W and 𝓙 is also continuous on W therefore we deduce the existence of a global minimizer v0 to 𝓙.

    Furthermore we note

    J(v0)J(v0+)+12qΩ(v0)2qdx+λΩ|v0|p(x)p(x)dx

    which implies v0 ≥ 0.

    Now we claim that v0 ≢ 0 in Ω. Since 𝓙(0) = 0, it is sufficient to prove the existence of W such that 𝓙() < 0. For that take = where ϕCc1(Ω) is nonnegative function such that ϕ ≢ 0 and t > 0 small enough.

    Since v0 is a global minimizer for the differentiable functional 𝓙, we have that v0 satisfies (4.3) i.e. v0 is a weak solution to (4.1). From Corollary A.1 we infer that v0L(Ω). Then by using Theorem A.2, we obtain, v0C1,α(Ω) for some α ∈ (0, 1).

    From (f2) and Lemma 3.3 (with condition (c2)), we obtain v0 > 0 and by Lemma 3.4 we get v0n<0 on ∂Ω. Therefore, v0 belongs to Cd0(Ω¯)+.

  • Step 2

    Contraction property (4.4)

    Let v1 and v2 two positive weak solutions of (4.1) such that v1, v2Cd0(Ω¯)+. For any ϕ, ΨW:

    Ωv12q1ϕdx+λΩ|v1|p(x)2v1.ϕdx=Ωh1v1q1ϕdx+λΩf(x,v1)ϕdx

    and

    Ωv22q1Ψdx+λΩ|v2|p(x)2v2.Ψdx=Ωh2v2q1Ψdx+λΩf(x,v2)Ψdx.

    Since v1,v2Cd0(Ω¯)+,ϕ=v1v2qv1q1+andΨ=v2v1qv2q1 are well-defined and belong to W. Subtracting the two above expressions and using (f3) together with Lemma 3.1 we obtain

    Ω((v1qv2q)+)2dxΩ(h1h2)(v1qv2q)+dx.

    Finally, applying the Hölder inequality we get (4.4).□

From Theorem 4.1, we deduce the accretivity of 𝓡:

Corollary 4.1

Let λ > 0, q ∈ (1, p], f : Ω × ℝ+ → ℝ+ satisfying (f1)-(f3) and h0L(Ω)+. Consider the following problem

u+λRu=h0(x)inΩ;u>0inΩ;u=0onΩ.(4.6)

Then there exists a unique distributional solution u ∈ 𝓓(𝓡) ∩ C1(Ω) of (4.6) i.e. ∀ϕCc1(Ω)

Ωu0ϕdx+λΩ|u01/q|p(x)2u01/q.(ϕu0(q1)/q)dx=Ωh0ϕdx+λΩf(x,u01/q)u0(q1)/qϕdx.

Moreover, if u1 and u2 are two distributional solutions of (4.6) in 𝓓(𝓡) ∩ C1(Ω) associated to h1 and h2 respectively, then the operator 𝓡 satisfies

(u1u2)+L2(u1u2+λ(Ru1Ru2))+L2.(4.7)

Proof

Define the energy functional 𝓔 on V˙+qL2(Ω) as 𝓔(u) = 𝓙(u1/q) where 𝓙 is defined in (4.5).

Let ϕCc1(Ω) and v0 is the global minimizer of (4.5) which is also the weak solution of (4.1) and u0 = v0q then there exists t0 = t0(ϕ) > 0 such that for t ∈ (–t0, t0), u0 + > 0. Hence we have

0E(u0+tϕ)E(u0)=12q(Ω(tϕ)2dx+Ω2tu0ϕdx)1qΩhtϕdx+λ(Ω|(u0+tϕ)1/q|p(x)p(x)dxΩ|u01/q|p(x)p(x)dx)λ(ΩF(x,(u0+tϕ)1/q)dxΩF(x,u01/q)dx).

Then divide by t and passing to the limits t → 0 we obtain u0 = v0q is the distributional solution of (4.6). Finally (4.7) and uniqueness follow from (4.4).□

We now generalize some above results for a larger class of potentials h0:

4.2 Further results for (4.1) and uniqueness

Theorem 4.2

Let λ > 0, f : Ω × ℝ+ → ℝ+ satisfying (f1)-(f3) and h0L2(Ω)+ and q ∈ (1, p]. Then there exists a positive weak solution vW of (4.1) in the sense of (4.3). Moreover assuming that h0 belongs to Lν(Ω) for some ν > max1,Np,vL(Ω).

Proof

Let hnCc1(Ω) such that hn ≥ 0 and hnh in L2(Ω). Define (vn) ⊂ C1,α(Ω) ∩ Cd0(Ω¯)+ as for a fixed n, vn is the unique positive weak solution of (4.1) with h0 = hn i.e. vn satisfies: for ϕW

Ωvn2q1ϕdx+λΩ|vn|p(x)2vn.ϕdx=Ωhnvnq1ϕdx+λΩf(x,vn)ϕdx.(4.8)

Since (ab)2q ≤ (aqbq)2 for any q ≥ 1, (4.4) implies for any n, p ∈ ℕ*

(vnvp)+L2q(hnhp)+L2q

thus we deduce that (vn) converges to vL2q(Ω).

We infer that the limit v does not depend on the choice of the sequence (hn). Indeed, consider nhn such that nh0 in L2(Ω) and n the positive solution of (4.1) corresponding to n which converges to .

Then, for any n ∈ ℕ, (4.4) implies

(vnqv~nq)+L2(hnh~n)+L2

and passing to the limit we get v and then by reversing the role of v and we obtain v = .

So define, for any n ∈ ℕ*, hn = min{h, n}. Thus (vn) is nondecreasing and for any n ∈ ℕ*, vnv a.e. in Ω which implies vv1 > 0 in Ω.

We choose ϕ = vn in (4.8). Applying the Hölder inequality and (4.2), we obtain

λΩ|vn|p(x)dxhnL2vnL2qq+λCϵvnL1+λϵvnLppC+λϵvnLpp.

Assume ∥∇vnLp(x) ≥ 1. Since W01,p(x)(Ω) ↪ Lp(Ω) and by (1.3) we deduce for some positive constant C > 0:

λΩ|vn|p(x)dxC+λϵCΩ|vn|p(x)dx.

Choosing ϵ small enough and gathering with the case ∥∇vnLp(x) ≤ 1, we conclude (vn) is uniformly bounded in W01,p(x)(Ω) and Lp(Ω). Hence vn converges weakly to v in W01,p(x)(Ω) and by monotonicity of (vn) strongly in Lp(Ω) and in L2q(Ω). Taking now ϕ = vnv in (4.8), from (4.2) with ϵ = 1 and by Hölder inequality

Ωf(x,vn)(vnv)dxCvnvL2q+vnLpp1vnvLp0

and

Ωhnvnq1(vnv)dx0andΩvn2q1(vnv)dx0.

Finally (4.8) becomes

Ω|vn|p(x)2vn.(vnv)dx0.

Then, since vnv in W01,p(x)(Ω)

Ω(|vn|p(x)2vn|v|p(x)2v).(vnv)dx0.

Lemma A.2 and Remark A.3 of [17] give the strong convergence of vn to v in W01,p(x)(Ω).

Since (vn2q1)and(hnvnq1) are uniformly bounded in L2q/(2q–1)(Ω) and by (4.2), f(x, vn) is uniformly bounded in L2q/q–1(Ω) and f(x, vn) → f(x, v) a.e. in Ω. Then by Lebesgue dominated convergence theorem we have (up to a subsequence), for ϕW

Ωvn2q1ϕdxΩv2q1ϕdx,Ωhnvnq1ϕdxΩhvq1ϕdx

and

Ωf(x,vn)ϕdxΩf(x,v)ϕdx.

Finally we pass to the limit in (4.8) and we obtain v is a weak solution of (4.1). To conclude corollary A.1 implies vL(Ω).□

Remark 4.2

Let v1, v2 are the weak solutions of (4.1) obtained by Theorem 4.2 corresponding to h1, h2L2(Ω)+, h1h2 respectively. Then

(v1qv2q)+L2(h1h2)+L2.

Remark 4.3

As in Step 1 of the proof of Theorem 4.1, we can alternatively prove the existence of a weak solution by global minimization method.

Under the hypothesis of Theorem 4.2 and with the help of Picone identity, the following theorem gives the uniqueness of the solution to (4.1).

Theorem 4.3

Let v, be respectively a subsolution and supersolution to (4.1) for hLp0(Ω), p0 ≥ 2, h ≥ 0 and f satisfies (f1) and (f3). Then v.

Proof

We have for any nonnegative ϕ, ΨW

Ωv2q1ϕdx+λΩ|v|p(x)2v.ϕdxΩhvq1ϕdx+λΩf(x,v)ϕdx

and

Ωv~2q1Ψdx+λΩ|v~|p(x)2v~.ΨdxΩhv~q1Ψdx+λΩf(x,v~)Ψdx.

Subtracting the above inequalities with test functions ϕ=((v+ϵ)q(v~+ϵ)q(v+ϵ)q1)+ and Ψ=((v~+ϵ)q(v+ϵ)q(v~+ϵ)q1)W for ϵ ∈ (0, 1), we obtain

{v>v~}(v2q1(v+ϵ)q1v~2q1(v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)dx+λ{v>v~}|(v+ϵ)|p(x)2(v+ϵ).((v+ϵ)q(v~+ϵ)q(v+ϵ)q1)dx+λ{v>v~}|(v~+ϵ)|p(x)2(v~+ϵ).((v~+ϵ)q(v+ϵ)q(v~+ϵ)q1)dx{v>v~}h(vq1(v+ϵ)q1v~q1(v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)dx+λ{v>v~}(f(x,v)(v+ϵ)q1f(x,v~)(v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)dx.(4.9)

Since v~v~+ϵvv+ϵ<1 in {v > }, then we obtain

(v2q1(v+ϵ)q1v~2q1(v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)=(vq(vv+ϵ)q1v~q(v~v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)vq((v+ϵ)q(v~+ϵ)q)vq(v+ϵ)qvq(v+1)q.

In the same fashion, we have

0h(vq1(v+ϵ)q1v~q1(v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)h(v+ϵ)qh(v+1)q.

Moreover, as ϵ → 0

(v2q1(v+ϵ)q1v~2q1(v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)(vqv~q)2

and

h(vq1(v+ϵ)q1v~q1(v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)0

a.e. in Ω. Then by Lebesgue dominated convergence theorem we have

{v>v~}(v2q1(v+ϵ)q1v~2q1(v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)dx{v>v~}(vqv~q)2dx(4.10)

and

{v>v~}h(vq1(v+ϵ)q1v~q1(v~+ϵ)q1)((v+ϵ)q(v~+ϵ)q)dx0.(4.11)

Then by using Fatou’s Lemma and (f1), we have

lim infϵ0{v>v~}f(x,v)(v+ϵ)q1(v~+ϵ)qdx{v>v~}f(x,v)vq1v~qdx,lim infϵ0{v>v~}f(x,v~)(v~+ϵ)q1(v+ϵ)qdx{v>v~}f(x,v~)v~q1vqdx(4.12)

and

{v>v~}f(x,v)(v+ϵ)dx{v>v~}f(x,v)vdx,{v>v~}f(x,v~)(v~+ϵ)dx{v>v~}f(x,v~)v~dx.(4.13)

By Lemma 3.1 we have,

{v>v~}|(v+ϵ)|p(x)2(v+ϵ).((v+ϵ)q(v~+ϵ)q(v+ϵ)q1)dx+{v>v~}|(v~+ϵ)|p(x)2(v~+ϵ).((v~+ϵ)q(v+ϵ)q(v~+ϵ)q1)dx0.(4.14)

Then plugging (4.10)-(4.14) and taking lim supϵ0 in (4.9), we get by (f3)

0{v>v~}(vqv~q)2dxλ{v>v~}(f(x,v)vq1f(x,v~)v~q1)(vqv~q)dx0.

It implies v.□

Corollary 4.2

Let λ > 0, f : Ω × ℝ+ → ℝ+ satisfying (f1)-(f3) and h0L2(Ω)+Lγ(Ω) where γ > max{1,Np}. Then there exists a unique positive distributional solution u ∈ 𝓓(𝓡) ∩ L(Ω) of (4.6) in the same sense as in Corollary 4.1.

Moreover if u1 and u2 are two positive distributional solutions of (4.6) for h1, h2L2(Ω)+ then 𝓡 satisfies

(u1u2)+L2(u1u2+λ(Ru1Ru2))+L2.(4.15)

Proof

Define the functional energy 𝓔 on V˙+qL2(Ω) as 𝓔(u) = 𝓙(u1/q) where 𝓙 is given by (4.5).

By Theorem 4.2, Remark 4.3 and Theorem 4.3, v0 is the unique positive solution of (4.1) and then unique global minimizer of 𝓙. We take u0 = v0q and proceed as the proof of Corollary 4.1 and we obtain u0 = v0q is a distributional solution of (4.6). Finally Remark 4.2 gives (4.15).□

4.3 Existence of a weak solution to (1.8)

In this section, in light of Remark 1.3, we consider the problem (1.9) and establish the existence of weak solution when v0Cd0(Ω¯)+W01,p(x)(Ω). Proof of Theorem 1.4: Let n* ∈ ℕ* and set Δt = T/n*. For 0 ≤ nn*, we define tn = t.

  • Step 1

    Approximation of h

    For n ∈ {1, 2, … n*}, we define for t ∈ [tn–1, tn) and xΩ

    hΔt(t,x)=hn(x)=def1Δttn1tnh(s,x)ds.

    Then by Jensen inequality,

    hΔtL2(QT)2=Δtn=1NhnL2(Ω)2=Δtn=1N1Δttn1tnh(s,x)dsL2(Ω)2n=1Ntn1tnh(s,.)L2(Ω)2dshL2(QT)2.

    Hence hΔtL2(QT) and hnL2(Ω) and let ϵ > 0, then there exists a function hϵC01(QT) such that hhϵL2(QT)<ϵ3.

    Hence,

    (hϵ)ΔthΔtL2(QT)0.

    Since ∥hϵ – (hϵ)ΔtL2(QT) → 0 as Δt → 0 then for small enough Δt we have

    hΔthL2(QT)(hϵ)ΔthΔtL2(QT)+hϵ(hϵ)ΔtL2(QT)+hhϵL2(QT)<ϵ.

    Hence hΔth in L2(QT).

  • Step 2

    Time discretization of (1.9)

    Define the following implicit Euler scheme and for n ≥ 1 , vn is the weak solution of

    (vnqvn1qΔt)vnq1Δp(x)vn=hnvnq1+f(x,vn) in Ω;vn>0 in Ω;vn=0 on Ω.(4.16)

    Note that the sequence (vn)n=1,2,…,n* is well-defined. Indeed for n = 1 the existence and the uniqueness of v1C1,α(Ω) ∩ Cd0(Ω¯)+ follows from Theorems 4.1 and 4.3 with h = Δth1 + v0qL(Ω)+. Hence by induction we obtain in the same way the existence and the uniqueness of the solution vn for any n = 2, 3, …, n* where vnC1,α(Ω) ∩ Cd0(Ω¯)+.

  • Step 3

    Existence of a subsolution and supersolution

    Now we construct a subsolution and a supersolution w and w of (4.16) such that for each n ∈ {0, 1, 2, …, n*}, vn satisfies 0 < wvnw.

    Rewrite (4.16) as

    vn2q1ΔtΔp(x)vn=Δthn+vn1qvnq1+Δtf(x,vn).(4.17)

    Then following arguments in the proof of Theorems 4.1 and 4.3, from Theorem A.2 and from Lemma 3.4, for any μ > 0 there exists a unique weak solution, wμC1,α(Ω) ∩ Cd0(Ω¯)+, to

    Δp(x)w=μ(h0wq1+f(x,w)) in Ω;w>0 in Ω;w=0 on Ω.(4.18)

    Let μ1 < μ2 and wμ1, wμ2 be weak solutions of (4.18). Then,

    Ω|wμ1|p(x)2wμ1.ϕdx=μ1Ω(h0wμ1q1+f(x,wμ1))ϕdx,Ω|wμ2|p(x)2wμ1.ψdx=μ2Ω(h0wμ2q1+f(x,wμ2))ψdx.

    Subtracting the last two equations with ϕ=(wμ1qwμ2q)+wμ1q1andψ=(wμ2qwμ1q)wμ2q1W01,p(x)(Ω) we obtain, by Lemma 3.1 and (f3), wμ1wμ2.

    Then by using Theorems A.2 and A.3, we can choose μ small enough such that ∥wμC1,α(Ω)Cμ0 for all μμ0 and ∥wμL(Ω) → 0 as μ → 0. Therefore {wμ : μμ0} is uniformly bounded and equicontinuous in C1(Ω) and by Arzela Ascoli theorem ∥wμC1(Ω) → 0 as μ → 0 up to a subsequence. Then by mean value theorem we can choose μ small enough such that there exists wC1,α(Ω) ∩ Cd0(Ω¯)+ such that 0 < w =def wμv0. Also w is the subsolution of (4.17) for n = 1 i.e.

    Ωw_2q1ϕdx+ΔtΩ|w_|p(x)2w_.ϕdxΔtΩ(h1w_q1+f(x,w_))ϕdx+Ωv0qw_q1ϕdx

    for all ϕW01,p(x)(Ω) and ϕ ≥ 0. We also recall v1 satisfies

    Ωv12q1ψdx+ΔtΩ|v1|p(x)2v1.ψdx=ΔtΩ(h1v1q1+f(x,v1))ψdx+Ωv0qv1q1ψdx

    for all ψW01,p(x)(Ω).

    By Theorem 4.3, we obtain, wv1 and then by induction a subsolution w such that 0 < wvn for all n = 0, 1, 2, …, n*.

    Now we construct a supersolution. For that, we consider the following problem:

    Δp(x)w=hLwq1+f(x,w)+K in Ω;w>0 in Ω;w=0 on Ω.(4.19)

    As above, there exists a unique weak solution to (4.19), wKC1(Ω) ∩ Cd0(Ω¯)+. Let wK be the unique weak solution of

    Δp(x)wK=K in Ω;wK=0 on Ω.(4.20)

    From Theorem A.3, wKCK1/(p+–1+ν) dist(x, ∂Ω) where ν ∈ (0, 1) and ∥wKL(Ω) → ∞ as K → ∞. Then by weak comparison principle we can choose K large enough such that there exists such that v0wK < w=def wK. We easily check that w is the supersolution of (4.17) for n = 1 i.e.

    Ωw¯2q1ϕdx+ΔtΩ|w¯|p(x)2w¯).ϕdxΔtΩ(h1w¯q1+K+f(x,w¯)ϕdx+Ωv0qw¯q1ϕdx

    for all ϕW01,p(x)(Ω) and ϕ ≥ 0. From Theorem 4.3, we get wv1 and then by induction we have wvn for all n ∈ {1, 2, … n*}.

  • Step 4

    Energy estimates

    Define the function for n = 1, … , n* and t ∈ [tn–1, tn)

    vΔt(t)=vnandv~Δt(t)=ttn1Δt(vnqvn1q)+vn1q

    which satisfies

    vΔtq1v~ΔttΔp(x)vΔt=f(x,vΔt)+hnvΔtq1.(4.21)

    Multiplying the equation (4.16) by vnqvn1qvnq1 and summing from n = 1 to n′ ≤ n*, we get

    n=1nΩΔt(vnqvn1qΔt)2dx+n=1nΩ|vn|p(x)2vn.(vnqvn1qvnq1)dx=n=1nΩhn(vnqvn1q)dx+n=1nΩf(x,vn)vnq1(vnqvn1q)dx.

    Then from Young inequality we have,

    n=1nΩΔt(vnqvn1qΔt)2dx+n=1nΩ|vn|p(x)2vn.(vnqvn1qvnq1)dxn=1nΔthnL22+14n=1nΩΔt(vnqvn1qΔt)2dx+n=1nΔtf(x,vn)vnq1L22+14n=1nΩΔt(vnqvn1qΔt)2dx

    i.e.

    12n=1nΩΔt(vnqvn1qΔt)2dx+n=1nΩ|vn|p(x)2vn.(vnqvn1qvnq1)dxn=1nΔthnL22+n=1nΔtf(x,vn)vnq1L22.

    Using wvnw, from (4.2) and qN2 + 1, we obtain

    Ωf(x,vn)vn(q1)2dxC1Ω1dist2(q1)(x,Ω)+dist2(pq)(x,Ω)dxC(4.22)

    where C is independent of n. Then by Step 1, we obtain

    (v~Δtt)is bounded inL2(QT)uniformly inΔt.(4.23)

    Now from Lemma 3.1, we have |vn|p(x)2vn.(vn1qvnq1)|vn1|q|vn|p(x)qqp(x)|vn1|p(x)+(p(x)q)p(x)|vn|p(x).

    Then we obtain for any n′ ≥ 1 n=1nΔthnL22+n=1nΔtf(x,vn)vnq1L22n=1nΩ|vn|p(x)2vn.(vnqvn1qvnq1)dxn=1nΩ|vn|p(x)dxΩqp(x)|vn1|p(x)dxΩ(p(x)q)p(x)|vn|p(x)dxqΩ|vn|p(x)p(x)dxqΩ|v0|p(x)p(x)dx

    which implies that (vΔt) is bounded in L(0,T;W01,p(x)(Ω))uniformly in Δt.(4.24)

    Since (v~Δt1/q)=1qζvn(ζ+(1ζ)(vn1vn)q)(1q)/q+(1ζ)vn1((1ζ)+ζ(vnvn1)q)(1q)/q

    where ζ=ttn1Δt, then we conclude that (v~Δt1/q) is bounded in L(0,T;W01,p(x)(Ω)) uniformly in Δt.(4.25)

    Since vnvn1 is uniformly bounded in L(Ω), vΔtv and v~Δt1/qv~ in L(0,T;W01,p(x)(Ω)). Furthermore using (4.23), we have supt(0,T)v~Δt1/qvΔtL2q(Ω)2qsupt(0,T)v~ΔtvΔtqL2(Ω)2Δt0 as Δt0.(4.26)

    It follows from (4.26) that v = . By mean value theorem and (4.23), we get that (Δt)Δt is equicontinuous in C(0, T; Lr(Ω)) for 1 < r ≤ 2. Thus using wqΔtwq together with the interpolation inequality ∥.∥r.α.21α, with 1r=α+1α2, we obtain that (Δt)Δt and (v~Δt1/q)Δt is equicontinuous in C(0, T; Lr(Ω)) for any 1 < r < +∞. Again using interpolation inequality and Sobolev embedding, we get as Δt → 0+ and up to a subsequence that for all r > 1 v~Δtvq in C(0,T;Lr(Ω)),(4.27)

    and vΔtv in L(0,T;Lr(Ω)).(4.28)

    From (4.23) and (4.27), we obtain v~Δttvqt in L2(QT).(4.29)

  • Step 5 :

    v satisfies (1.10)

    Multiplying (4.21) by (vΔtv) and integrating by parts, we get 0TΩvΔtq1v~Δtt(vΔtv)dxdt+0TΩ|vΔt|p(x)2vΔt.(vΔtv)dxdt=0TΩf(x,vΔt)(vΔtv)dxdt+0TΩhnvΔtq1(vΔtv)dxdt.

    From (4.28) and (4.29) , we have 0TΩvΔtq1v~Δtt(vΔtv)dxdt+0TΩhnvΔtq1(vΔtv)dxdt=oΔt(1)

    and from (4.24), (4.25), (4.28) and Lebesgue Dominated convergence theorem, 0TΩf(x,vΔt)(vΔtv)dx=oΔt(1).

    Then we obtain 0TΩ|vΔt|p(x)2vΔt.(vΔtv)dx0 as Δt0+.

    Then from [Step 4, Proof of Theorem 1.1, [12]] and from classical compactness argument we get |vΔt|p(x)2vΔt|v|p(x)2v in (Lp(x)/(p(x)1)(QT))N.(4.30)

    From (4.26) and (4.27) we have, vΔtq1vq1L2(QT)vΔtq1vq1L(0,T;L2)vΔtq1vq1L(0,T;L2qq1)vΔtqvqL(0,T;L2)vΔtqv~ΔtL(0,T;L2)+v~ΔtvqL(0,T;L2)0(4.31)

    as Δt → 0. By Hölder inequality we have for ϕCc(QT) 0TΩ(vΔtq1v~Δttvqtvq1)ϕdx=0TΩvΔtq1(v~Δttvqt)ϕdx+0TΩvqt(vΔtq1vq1)ϕdxvΔtq1ϕL2(QT)v~ΔttvqtL2(QT)+vΔtq1vq1L2(QT)ϕv~ΔttL2(QT)

    and 0TΩ(hnvΔtq1hvq1)ϕdx=0TΩhn(vΔtq1vq1)ϕdx+0TΩ(hnh)vq1ϕdxhnϕL2(QT)vΔtq1vq1L2(QT)+vq1ϕL2(QT)hnhL2(QT).

    Then from (4.23), (4.28), (4.29), (4.31) and Step 1 we obtain 0TΩ(vΔtq1v~Δttvqtvq1)ϕdx0,0TΩ(hnvΔtq1hvq1)ϕdx0 as Δt0.(4.32)

    From (4.28) we have f(x, vΔt) → f(x, v) pointwise and from (4.24) together with (4.25) we have Ω f(x, vΔt) ϕ dx is bounded uniformly in Δt. Then by Lebesgue dominated convergence theorem we have 0TΩ(f(x,vΔt)f(x,v))ϕdx0 as Δt0.(4.33)

    Then finally gathering (4.30), (4.32) and (4.33), we conclude by passing to the limits in equation (4.21) that v is weak solution of (1.9). □

Remark 4.4

For q > N2+1, if f satisfies lims0+f(x,s)sα=0 where α > q − 1 − N2 then Theorem 1.4 holds. Since wvnw then (4.22) is in this case Ωf(x,vn)vnq12dxC1Ωw¯2αdist2(q1)(x,Ω)dx+C2C

where C is independent of n.

Remark 4.5

All the results in Section 4.1, Section 2 and Theorem 1.4 hold if we replace the assumption (f2) by hc > 0.

Proof of Theorem 1.5

For a given function g, let g2+=def[g]+L2(Ω). For z ∈ 𝓓(𝓡) and r, kL(QT)+ satisfying assumptions in Theorem 1.5, set ϕ(t,s)=r(t)k(s)2+(t,s)[0,T]×[0,T],

for t ∈ [−T, T] b(t,r,k)=u0qz2++v0qz2++|t|Rz2++0t+r(τ)2+dτ+0tk(τ)2+dτ

and ψ(t,s)=b(ts,r,k)+0sϕ(ts+τ,τ)dτ if 0stT0tϕ(τ,st+τ)dτ if 0tsT

is a solution of ψt(t,s)+ψs(t,s)=ϕ(t,s) on (t,s)[0,T]×[0,T];ψ(t,0)=b(t,r,k) on t[0,T];ψ(0,s)=b(s,r,k) on s[0,T].(4.34)

Define the following iterative scheme, u0 = u0q and for n ≥ 1 , un is the solution of unun1Δt+Run=hn in Ω;un=0 on Ω.(4.35)

Note that the sequence {un}n = 1, 2, … , N is well defined. Indeed for n = 1 the existence and the uniqueness of u1 ∈ 𝓓(𝓡) follows from Corollary 4.1 with h = Δt h1 + u0L(Ω)+ and λ = Δt. Hence by induction we obtain in the same way the existence of the solution un for any n = 2, 3, … , N where un ∈ 𝓓(𝓡).

Moreover let denote by (uϵn) the solution of (4.35) with Δt = ϵ , h = r, rn=1ϵ(n1)ϵnϵr(τ,.)dτ and (uηm) the solution of (4.35) with Δt = η , h = k, km=1η(m1)ηmηk(τ,.)dτ respectively i.e we have uϵnuϵn1ϵ+Ruϵn=rn;uηmuηm1η+Ruηm=km.(4.36)

For (n, m) ∈ ℕ*, multiplying the equation in (4.36) by ϵηϵ+η and then subtracting the two expressions we get, ηη+ϵ(uϵnuϵn1)+ηϵη+ϵ(RuϵnRuηm)ϵη+ϵ(uηmuηm1)=ηϵη+ϵ(rnkm).

Then we infer that uϵnuηm+ϵηϵ+η(RuϵnRuηm)=ϵηϵ+η(rnkm)+ηϵ+η(uϵn1uηm)+ϵϵ+η(uϵnuηm1).

Let Φn,mϵ,η=uϵnuηm2+ and since 𝓡 satisfies (4.15) and setting λ=ϵηϵ+η, we get Φn,mϵ,η=uϵnuηm2+uϵnuηm+ϵηϵ+η(RuϵnRuηm)2+ϵηϵ+ηrnkm2++ηϵ+ηuϵn1uηm2++ϵϵ+ηuϵnuηm12+.

Then by elementary calculations, we get Φn,0ϵ,η=uϵnuη2+b(tn,rϵ,kη)

and Φ0,mϵ,ηb(sm,rϵ,kη).

Then by using above computations we get, Φn,mϵ,ηψn,mϵ,η where ψn,mϵ,η satisfies ψn,mϵη=ϵηϵ+η(rnkm)2++ηϵ+ηψn1,mϵ,η2++ϵϵ+ηψn,m1ϵ,η2+

and ψn,0ϵ,η=b(tn,rϵ,kη) and ψ0,mϵ,η=b(sm,rϵ,kη).

For (t, s) ∈ (tn−1, tn) × (sm−1, sm), set ϕϵ, η (t, s) = ∥rϵ(t) − kη(s)∥2+, ψϵ,η=ψn,mϵ,η,bϵ,η(t,r,k)=b(tn,rϵ,kη),bϵ,η(s,r,k)=b(sm,rϵ,kη).

Then by elementary calculations ψϵ, η satisfies the following discrete version of (4.34), ψϵ,η(t,s)ψϵ,η(tϵ,s)ϵ+ψϵ,η(t,s)ψϵ,η(t,sη)η=ϕϵ,η(t,s);ψϵ,η(t,0)=bϵ,η(t,r,k);ψϵ,η(0,s)=bϵ,η(s,r,k).

Since rϵr in L2(QT) then bϵ, η (., r, k) → b(., r, k) in L([0, T]) and ϕϵ, ηϕ in L([0, T] ×[0, T]) and we deduce that ρϵ, η = ∥ψϵ, ηψL([0, T] ×[0, T]) → 0 (for more details see for instance [[20], Chapter 4, Lemma 4.3, page 136] and [[20], Chapter 4, Proof of Theorem 4.1, page 138]). Therefore, uϵ(t)uη(s)2+=Φϵ,η(t,s)ψϵ,η(t,s)ψ(t,s)+ρϵ,η.

Since uϵ(t)=vϵq(t) and uη(t)=vηq(t), we obtain vϵq(t)vηq(s)2+=Φϵ,η(t,s)ψϵ,η(t,s)ψ(t,s)+ρϵ,η.(4.37)

From Theorem 1.4, vϵq and vηq satisfies 0 < w < vϵ, vη < w where w, w are subsolution and supersolution defined in (4.18) and (4.19) and vϵqv1q and vηqv2q a.e. in Ω where v1 and v2 are weak solutions of (1.9) with initial data u0, v0 respectively. Since vϵqv1q and vηqv2q in L(0, T;L2(Ω)) and passing to the limit in (4.37) as ϵ, η → 0 with t = s we get v1q(t)v2q(t)2+v1q(t)vϵq(t)2++vηq(t)v2q(t)2++vϵq(t)vηq(t)2+u0qz2++v0qz2++0tr(γ)k(γ)2+dγ.

Then (1.11) follows since we can choose z arbitrary close to v0q and with r = h, k = g. □

5 An application to nonhomogeneous operators

In this final section, we prove Theorem 1.6. To this aim, we first study the properties of a related energy functional. Let m ≥ 1 and K : Ω × ℝN → ℝ+ be a continuous differentiable function which satisfies the following conditions:

  • (k1)

    KC1(Ω × ℝN) ∩ C2(Ω × ℝN ∖ {0}).

  • (k2)

    Ellipticity condition: ∃ k1 ≥ 0 and y ∈ (0, ∞) such that i,j=1N2Kξiξj(x,ξ)ηiηjy(k1+|ξ|)m2|η|2.

  • (k3)

    Growth condition: ∃ k2 ≥ 0 and Γ ∈ (0, ∞) such that i,j=1N2Kξiξj(x,ξ)Γ(k2+|ξ|)m2

    for all ξ ∈ ℝN ∖ {0} and η ∈ ℝN.

Remark 5.1

From the assumption (k2), it follows that K is strictly convex and from (k1)-(k3) there exists some positive constant y1 and y2 with 0 < y1y2 < +∞ and some nonnegative constants Γ1 and Γ2 such that y1|ξ|mΓ1K(x,ξ)y2|ξ|m+Γ2

for xΩ and ξ ∈ ℝN ∖ {0}.

Consider the associated functional 𝓙m defined by Jm(u)=defΩ|u|p(x)p(x)K(x,uu)p(x)mdx.

for any positive function uW01,p(x)(Ω). Now we extend Lemma 2.4 in [21] as follows:

Theorem 5.1

Let K : Ω × ℝN → ℝ+ satisfying (k1)-(k3) for some m ∈ [1, p]. Then, the function 𝓔 : V˙+mL(Ω) → ℝ+, defined by E(u)=defJm(u1/m), is ray-strictly convex (even strictly convex if p(⋅)≢ m).

Proof

We observe that for uV˙+mL(Ω) E(u)=Ω1p(x)uKx,umup(x)mdx.

Therefore, since for 1 ≤ mp, ttp(x)/m is convex in ℝ+ (even strictly convex if p(x) > m) it is enough to prove that V˙+muuK(x,umu)

is ray-strictly convex. To achieve this goal, let θ ∈ (0, 1) and u1, u2V˙+m then by using the strict convexity of K we obtain, for xΩ ((1θ)u1+θu2)K(x,(1θ)u1+θu2m((1θ)u1+θu2))=((1θ)u1+θu2)K(x,(1θ)u1((1θ)u1+θu2)u1mu1+θu2((1θ)u1+θu2)u2mu2)((1θ)u1+θu2)((1θ)u1((1θ)u1+θu2)K(x,u1mu1)+θu2((1θ)u1+θu2)K(x,u2mu2))=(1θ)u1K(x,u1mu1)+θu2K(x,u2mu2).

The above inequality is always strict unless u1u1=u2u2, i.e. u1/u2 ≡ Const. □

Proof of Theorem 1.6

Consider the functional 𝓙ϵ : W01,p(x) (Ω) → ℝ , defined by Jϵ(u)=Ω(|u|2+ϵu2)p(x)/2p(x)dxΩG(x,u)dx

where the potential G(x, t) defined as G(x,t)=0tg(x,s)ds if 0t<;0 if <t<0.

Assumptions (f1), () and Remark 5.1 ensure that 𝓙ϵ is well defined, coercive and continuous. Then there exists at least one global minimizer of 𝓙ϵ on W01,p(x)(Ω), say u0. We can easily prove that u0 is nonnegative and nontrivial.

Since 𝓙ϵ is differentiable, we deduce that u0 is a weak solution of (1.12). Now from Theorems A.1 and A.2 in Appendix A, we obtain that any weak solution u to (1.12) belongs to C1, α(Ω) for some α ∈ (0, 1) and u > 0 in Ω and un < 0 on ∂Ω. Therefore any weak solution belongs to Cd0(Ω¯)+.

Now we prove that u0 is the unique weak solution to (1.12). Let W : V˙+m → ℝ defined by W(u)=Jϵ(u1/m)=Ω(|(u1/m)|2+ϵ(u1/m)2)p(x)/2p(x)dxΩG(x,u1/m)dx.

The assumption () together with Theorem 5.1 with K(x, ξ) = (ϵ + |ξ|2)m/2 imply that W is strictly convex.

Let u1 a weak solution to (1.12). Then setting v0=defu0m,v1=defu1mV˙+m and t ∈ [0, 1], we define ξ(t)=def Jϵ(((1 − t)v0+ t v1)1/m). Since u0 and u1 belong to Cd0(Ω¯)+, ξ is differentiable in [0, 1]. From the convexity of 𝓔, we have for any t ∈ [0, 1] ξ(0)ξ(t)ξ(1).(5.1)

Since u0 and u1 are weak solutions to (1.12), ξ′(0) = ξ′(1) = 0 and from (5.1) we get that ξ is constant which contradicts the strict convexity of 𝓔 unless u0u1. □

A Appendix

In this section, we recall the following regularity of weak solutions of quasilinear elliptic differential equation divA(x,u,Du)+B(x,u,Du)=0 on Ω;u=0 on Ω.(A.1)

Now we recall the boundedness and C0, α(Ω) regularity results of weak solutions of (A.1) satisfying the following growth conditions: A(x,u,z)za0|z|p(x)b|u|r(x)c;|A(x,u,z)|a1|z|p(x)1+b|u|σ(x)+c;|B(x,u,z)|a2|z|α(x)+b|u|r(x)1+c(A.2)

where a0, a1, a2, b, c are positive constants and p* is the Sobolev embedding exponent of p and p(x)r(x)<p(x),σ(x)=p(x)1p(x)r(x) and α(x)=r(x)1r(x)p(x).(A.3)

Theorem A.1

([11], Theorem 4.1 and Theorem 4.4) Let (A.2)-(A.3) hold and p ∈ 𝓟log(Ω). If uW1, p(x)(Ω) is a weak solution of (A.1), then uC0, α({Ω).

Theorem A.2 below ensures C1, α(Ω) regularity to weak solutions of (A.1) under the additional assumptions on p, A and B:

Assumptions (Ak): A = (A1, A2, …, An) ∈ C(Ω × ℝ × ℝN, ℝN). For every (x, u) ∈ Ω × ℝ, A(x, u, .) ∈ C1 (ℝN ∖ {0}, ℝN), there exist a nonnegative constants k1, k2, k3 ≥ 0, a nonincreasing continuous function λ : [0, ∞) → (0, ∞) and a nondecreasing continuous function Λ : [0, ∞) → (0, ∞) such that for all x, x1, x2Ω, u, u1, u2 ∈ ℝ , η ∈ ℝN ∖ {0} and ξ = (ξ1, ξ2, …, ξn) ∈ ℝN, the following conditions are satisfied A(x,u,0)=0,i,jAj(x,u,η)ηi(x,u,η)ξiξjλ(|u|)(k1+|η|2)(p(x)2)/2|ξ|2,i,j|Aj(x,u,η)ηi(x,u,η)|Λ(|u|)(k2+|η|2)(p(x)2)/2 and |A(x1,u1,η)A(x2,u2,η)|Λ(max{|u1|,|u2|})(|x1x2|β1+|u1u2|β2)×[(k+|η|2)(p(x1)2)/2+(k+|η|2)(p(x2)2)/2]|η|(1+|log(k3+|η|2)|).

Assumption (B): B :Ω × ℝ × ℝN → ℝ, the function B(x, u, η) is measurable in x and is continuous in (u , η), and |B(x,u,η)|Λ(|u|)(1+|η|p(x)),(x,u,η)Ω¯×R×RN.

Theorem A.2

(10], Theorem 1.2) Let assumptions (Ak), (B) hold. Assume p belongs to C0, β(Ω), for some β ∈ (0, 1). Suppose that Ω satisfy (Ω). If uW01,p(x)(Ω) ∩ L(Ω) is a weak solution of (A.1), then uC1, α(Ω) where α ∈ (0, 1) anduC1, α(Ω) depends upon p, p+, λ(M), Λ(M), β1, β2, M, Ω where M=defuL(Ω).

In the next theorem, we recall some results contained in Lemma 2.1 of [22] and Lemma 3.2 of [12]. Set ϱ = p2|Ω|1/NC0 where C0 is the best embedding constant of W01,1(Ω)LNN1(Ω).

Theorem A.3

Let K > 0 and wKW01,p(x)(Ω) ∩ C1, α(Ω) be the weak solution of (4.20).

Then for Kϱ, ∥wKL(Ω)C1 K1/(p−1), wK(x) ≥ C2 K1/(p+−1+ ς) dist(x, ∂Ω) where ς ∈ (0, 1) and for K < ϱ, ∥wKL(Ω)C3 K1/(p+−1) where C1, C2 and C3 depends upon p+, p, N, Ω. Moreover if K1 < K2 then wK1wK2.

Next we prove a slight extension of Proposition A.2 in [12].

Proposition A.1

Let pC(Ω) and q ∈ (1, p]. Assume uW satisfying for any ΨW: Ω|u|p(x)2u.Ψdx=Ωhuq1Ψdx(A.4)

where hL2(Ω) ∩ Lr(Ω) with r>max{1,Np}. Then uL(Ω).

First we prove a regularity lemma.

Lemma A.1

Let uW01,p(Ω) satisfying for any BR, R < R0, and for all σ ∈ (0,1), and any kk0 > 0

Ak,σR|u|pdxCAk,RukR(1σ)pdx+kα|Ak,R|+|Ak,R|pp+ε+kβ|Ak,R|pp+ε+Ak,RukR(1σ)pdxpp|Ak,R|δ

where Ak,R = {xBRΩu(x) > k}, 0 < α < p=NpNp, β ∈ (1, p] and ε, δ > 0. Then uL(Ω).

Proof

A similar result exists in [23] or in [17] without the term kβ|Ak,R|pp+ε. For the reader’s convenience, we include the complete proof.

Let x0Ω, BR the ball centred in x0. We define KR=defBRΩ and we set

rj=R2+R2j+1,r~j=rj+rj+12 and kj=k112j+1 for any jN.

Define also

Ij=Akj,rj|u(x)kj|pdxandφ(t)=1if0t12,0ift34

satisfying φC1([0, +∞)) and 0 ≤ φ ≤ 1. We set φj(x)=φ2j+1R(|x|R2). Hence φj = 1 on Brj+1 and φj = 0 on ℝNBj+1.

We have

Ij+1=Akj+1,rj+1|u(x)kj+1|pdx=Akj+1,rj+1|u(x)kj+1|pφj(x)pdxKR(u(x)kj+1)+φj(x))pdx.

Since uW01,p(Ω), (ukj+1)+ φjW01,p(KR),

Ij+1KR|((ukj+1)+φj)|pdxppAkj+1,r~j|u|pdx+Akj+1,r~j(ukj+1)pdxpp

where we use the notation fg in the sense there exists a constant c > 0 such that fcg. Since j < rj, we have

Ij+12jpAkj+1,rj|ukj+1|pdx+kj+1α|Akj+1,rj|+|Akj+1,rj|pp+ε+kj+1β|Akj+1,rj|pp+ε+2jpAkj+1,rj|ukj+1|pdxpp|Akj+1,rj|δ+Akj+1,rj|ukj+1|pdxpp.(A.5)

Moreover, for any j, kjkj+1, this implies

IjAkj+1,rj|ukj|pdxAkj+1,rj|kjkj+1|pdx=|Akj+1,rjkj+1kj|p.

Then, for any k > k0 and j ∈ ℕ

|Akj+1,rj|+kj+1p|Akj+1,rj|2jpIj

where the constant in the notation depends only on k0, p and α. From the previous inequality, we deduce

kj+1β|Akj+1,rj|pp+εkj+1p+εp|Akj+1,rj|pp+ε2j(p+εp)Ijpp+ε.

Replacing in (A.5), we obtain

Ij+1(2jpIj+2j(p+εp)Ijpp+ε+2j(p+δp)Ijpp+δ)pp.(A.6)

Setting M=ppmax{p,p+εp,p+δp} and θ=min{1pp,ε,δ} and noting

IjKR(|ukj|+)pdxKR|u|puW01,pp,

(A.6) becomes

Ij+12jMIj1+θpp

where the constant depends on uW01,p,k0,α and p. We conclude with Lemma 4.7 in Chapter 2 of [24].

For this it suffices to prove that I0 is small enough. Indeed uLp*(Ω) implies

I0=Ak2,R|uk2|pdx0 as k.

Hence for k large enough, I0C1η(2M)1η2 with η=θpp. Thus Ij converges to 0 as j → +∞ and

Ak,R2|uk|pdx=0.

We deduce that uk on KR2. In the same way, we prove that −uk on KR2.

Since Ω is compact, we conclude that uL(Ω).□

Proof of Proposition A.1:

We follow the idea of the proof of Theorem 4.1 in [11].

Let x0Ω, BR the ball of radius R centered in x0 and KR=defΩBR. We define

p+=defmaxKRp(x)andp=defminKRp(x)

and we choose R small enough such that p+ < (p)* where

(p)=defNpNpif p<N,p++1if pN.

Fix (s, t) ∈ (R+)2, t < s < R then KtKsKR. Define φC(Ω), 0 ≤ φ ≤ 1 such that

φ=1in Bt,0in RNBs

satisfying ∣∇φ∣ ≲ 1/(st). Let k ≥ 1, using the same notations as previously Ak,λ = {yKλu(y) > k} and taking Ψ = φp+(uk)+W01,p(x)(Ω) in (A.4), we obtain

Ak,s|u|p(x)φp+dx+p+Ak,s|u|p(x)2uφφp+1(uk)+dx=Ak,shuq1φp+(uk)dx.(A.7)

Hence by Young inequality, for ϵ > 0, we have

p+Ak,s|u|p(x)2uφφp+1(uk)dxεAk,s|u|p(x)φ(p+1)p(x)p(x)1dx+cε1Ak,s(uk)p(x)|φ|p(x)dx.

Since ∣∇φ∣ ≤ c/(st) and for any xKR, p+(p+1)p(x)p(x)1, we have φ(p+1)p(x)p(x)1φp+. This implies

p+Ak,s|u|p(x)2u.φφp+1(uk)dxεAk,s|u|p(x)φp+dx+cε1Ak,sukstp(x)dx.(A.8)

Using Hölder inequality, we estimate the right-hand side of (A.7) as follows:

Ak,shuq1φp+(uk)dxhLrAk,sur(q1)r1(uk)rr1dxr1r.

Since r>Np, we have (p)pr1r>1, applying once again the Hölder inequality and the Young inequality, we obtain

Ak,shuq1φp+(uk)dxAk,suq(p)pdx+Ak,s(uk)q(p)pdxp(p)|Ak,s|δ

where δ=r1rp(p)>0.

Set Ak,s,t = {xAk,su(x) − k > st} and its complement as Ak,s,tc. Now we split the integrals in the right-hand side of (A.9) as follows:

Ak,s,tukstq(p)p(st)q(p)pdx+Ak,s,tcukstq(p)p(st)q(p)pdxAk,sukst(p)dx+|Ak,s|=defI(A.9)

since q < p and we also have

Ak,suq(p)pdxAk,s(uk)q(p)p+kq(p)pdxI+kq(p)p|Ak,s|.

In the same way, the second term in the right-hand side of (A.8) can be estimated as follows:

Ak,sAk,s,tukstp(x)dx+Ak,sAk,s,tcukstp(x)dxI.(A.10)

Finally plugging (A.8)-(A.10), we obtain for ε small enough

Ak,s|u|p(x)φp+dxI+|Ak,s|δ(I+kq(p)p|Ak,s|)p(p)

where the constant depends on p, R and ε. Moreover we have

(I+kq(p)p|Ak,s|)p(p)Ak,sukst(p)dxp(p)+|Ak,s|p(p)+kq|Ak,s|p(p).

To conclude, using the Young inequality, we obtain the following estimate:

Ak,t|u|pdxAk,s|u|p(x)φp+dxAk,sukst(p)dx+2|Ak,s|+(1+kq)|Ak,s|p(p)+δ+|Ak,s|δAk,sukst(p)dxp(p).

By Lemma A.1, we deduce that u bounded in Ω.□

Combining Theorem 4.1 of [11] and Proposition A.1, we have the following corollary:

Corollary A.1

Let pC(Ω̄) and q ∈ (1, p]. Assume uW and nonnegative satisfying for any ΨW, Ψ ≥ 0,

Ωu2q1Ψdx+Ω|u|p(x)2uΨdxΩ(f(x,u)+huq1)Ψdx

where f verifies for any (x, t) ∈ Ω × ℝ+, ∣f(x, t)∣ ≤ c1 + c2ts(x)−1 with sC(Ω) such that for any xΩ, 1 < s(x) < p*(x) and hL2(Ω) ∩ Lr(Ω) with r>max{1,Np}. Then uL(Ω).

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About the article

Received: 2018-10-27

Accepted: 2018-11-29

Published Online: 2019-05-16

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 327–360, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0003.

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© 2020 R. Arora et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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